Distributed-element circuit

841: 1355:, invented in 1952, became a commercial rival of stripline; however, planar formats did not start to become widely used in microwave applications until better dielectric materials became available for the substrates in the 1960s. Another structure which had to wait for better materials was the dielectric resonator. Its advantages (compact size and high quality) were first pointed out by R. D. Richtmeyer in 1939, but materials with good temperature stability were not developed until the 1970s. Dielectric resonator filters are now common in waveguide and transmission line filters. 895: 283: 556: 224:). A common rule of thumb amongst engineers is to change from the lumped to the distributed model when distances involved are more than one-tenth of a wavelength (a 36° phase change). The lumped model completely fails at one-quarter wavelength (a 90° phase change), with not only the value, but the nature of the component not being as predicted. Due to this dependence on wavelength, the distributed-element model is used mostly at higher frequencies; at low frequencies, distributed-element components are too bulky. Distributed designs are feasible above 1181: 940: 488: 152: 1014: 599: 1258: 381:, used for telephone lines and Internet connections. It is not often used for distributed-element circuits because the frequencies used are lower than the point where distributed-element designs become advantageous. However, designers frequently begin with a lumped-element design and convert it to an open-wire distributed-element design. Open wire is a pair of parallel uninsulated conductors used, for instance, for 705: 20: 413: 427:, a centre conductor surrounded by an insulated shielding conductor, is widely used for interconnecting units of microwave equipment and for longer-distance transmissions. Although coaxial distributed-element devices were commonly manufactured during the second half of the 20th century, they have been replaced in many applications by planar forms due to cost and size considerations. Air- 579:. The coupling can be direct or indirect. In indirect coupling, the two lines are run closely together for a distance with no screening between them. The strength of the coupling depends on the distance between the lines and the cross-section presented to the other line. In direct coupling, branch lines directly connect the two main lines together at intervals. 716:-like curves as a circuit component is an emerging field in distributed-element circuits. Fractals have been used to make resonators for filters and antennae. One of the benefits of using fractals is their space-filling property, making them smaller than other designs. Other advantages include the ability to produce 298:, a particularly simple form to model. The cross-sectional dimensions of the line are unvarying along its length, and are small compared to the signal wavelength; thus, only distribution along the length of the line need be considered. Such an element of a distributed circuit is entirely characterised by its length and 1226:), with all their inputs connected via one transmission line and all their outputs via another transmission line. The lengths of the two lines must be equal between each transistor for the circuit to work correctly, and each transistor adds to the output of the amplifier. This is different from a conventional 1235:

amplifier, the overall bandwidth is the same as the bandwidth of a single stage. Distributed amplifiers are used when a single large transistor (or a complex, multi-transistor amplifier) would be too large to treat as a lumped component; the linking transmission lines separate the individual transistors.

1291:

was the first to investigate the possibility of distributed-element circuits, and filed a patent in 1927 for a coaxial filter designed by this method. Mason and Sykes published the definitive paper on the method in 1937. Mason was also the first to suggest a distributed-element acoustic filter in his

1171:

for an ideal three-port circulator, showing that circulators are non-reciprocal by definition. It follows that it is impossible to build a circulator from standard passive components (lumped or distributed). The presence of a ferrite, or some other non-reciprocal material or system, is essential for

879:

Impedance matching for narrow-band applications is frequently achieved with a single matching stub. However, for wide-band applications the impedance-matching network assumes a filter-like design. The designer prescribes a required frequency response, and designs a filter with that response. The only

627:

as a distributed-element circuit. The quarter-wave transformers are alternated with a distributed-element resonator to achieve this. However, this is now a dated design; more compact inverters, such as the impedance step, are used instead. An impedance step is the discontinuity formed at the junction

614:

Cascaded lines are lengths of transmission line where the output of one line is connected to the input of the next. Multiple cascaded lines of different characteristic impedances can be used to construct a filter or a wide-band impedance matching network. This is called a stepped impedance structure.

506:

over conducting lines, but their relative expense and bulk means that microstrip is often preferred. Waveguide mostly finds uses in high-end products, such as high-power military radars and the upper microwave bands (where planar formats are too lossy). Waveguide becomes bulkier with lower frequency,

73:

Conventional circuits consist of individual components manufactured separately then connected together with a conducting medium. Distributed-element circuits are built by forming the medium itself into specific patterns. A major advantage of distributed-element circuits is that they can be produced

1382:

in 1957, should be considered the first fractal antenna. However, its self-similar nature, and hence its relation to fractals was missed at the time. It is still not usually classed as a fractal antenna. Cohen was the first to explicitly identify the class of fractal antennae after being inspired

776:

A taper is a transmission line with a gradual change in cross-section. It can be considered the limiting case of the stepped impedance structure with an infinite number of steps. Tapers are a simple way of joining two transmission lines of different characteristic impedances. Using tapers greatly

662:

A dielectric resonator is a piece of dielectric material exposed to electromagnetic waves. It is most often in the form of a cylinder or thick disc. Although cavity resonators can be filled with dielectric, the essential difference is that in cavity resonators the electromagnetic field is entirely

358:

and resistive losses in lumped components are greater with increasing frequency as a proportion of the nominal value of the lumped-element impedance. In some cases, designers may choose a distributed-element design (even if lumped components are available at that frequency) to benefit from improved

1339:

and was responsible for many filter designs. Matthaei first described the interdigital filter and the combline filter. The group's work was published in a landmark 1964 book covering the state of distributed-element circuit design at that time, which remained a major reference work for many years.

1311:

which operated in the microwave band and resulted in radar equipment small enough to install in aircraft. A surge in distributed-element filter development followed, filters being an essential component of radars. The signal loss in coaxial components led to the first widespread use of waveguide,

1030:

A circulator is usually a three- or four-port device in which power entering one port is transferred to the next port in rotation, as if round a circle. Power can flow in only one direction around the circle (clockwise or anticlockwise), and no power is transferred to any of the other ports. Most

923:

when the coupling factor is high. A power combiner is simply a power splitter used in reverse. In distributed-element implementations using coupled lines, indirectly coupled lines are more suitable for low-coupling directional couplers; directly coupled branch line couplers are more suitable for

524:

band (below microwave frequencies), where waveguides might otherwise be used. Mechanical circuits can also be implemented, in whole or in part, as distributed-element circuits. The frequency at which the transition to distributed-element design becomes feasible (or necessary) is much lower with

856:

Filters are a large percentage of circuits constructed with distributed elements. A wide range of structures are used for constructing them, including stubs, coupled lines and cascaded lines. Variations include interdigital filters, combline filters and hairpin filters. More-recent developments

981:

Another use for a hybrid coupler is to produce the sum and difference of two signals. In the illustration, two input signals are fed into the ports marked 1 and 2. The sum of the two signals appears at the port marked Σ, and the difference at the port marked Δ. In addition to their uses as

353:

Distributed-element circuits are cheap and easy to manufacture in some formats, but take up more space than lumped-element circuits. This is problematic in mobile devices (especially hand-held ones), where space is at a premium. If the operating frequencies are not too high, the designer may

1234:

is multiplied by the gain of each stage. Although a distributed amplifier has lower gain than a conventional amplifier with the same number of transistors, it has significantly greater bandwidth. In a conventional amplifier, the bandwidth is reduced by each additional stage; in a distributed

1292:

1927 doctoral thesis, and a distributed-element mechanical filter in a patent filed in 1941. Mason's work was concerned with the coaxial form and other conducting wires, although much of it could also be adapted for waveguide. The acoustic work had come first, and Mason's colleagues in the

663:

contained within the cavity walls. A dielectric resonator has some field in the surrounding space. This can lead to undesirable coupling with other components. The major advantage of dielectric resonators is that they are considerably smaller than the equivalent air-filled cavity.

781:

between lines in different media, especially different forms of planar media. Tapers commonly change shape linearly, but a variety of other profiles may be used. The profile that achieves a specified match in the shortest length is known as a Klopfenstein taper and is based on the

902:

A directional coupler is a four-port device which couples power flowing in one direction from one path to another. Two of the ports are the input and output ports of the main line. A portion of the power entering the input port is coupled to a third port, known as the

313:

Commensurate line circuits are important because a design theory for producing them exists; no general theory exists for circuits consisting of arbitrary lengths of transmission line (or any arbitrary shapes). Although an arbitrary shape can be analysed with

1303:, there was little demand for distributed-element circuits; the frequencies used for radio transmissions were lower than the point at which distributed elements became advantageous. Lower frequencies had a greater range, a primary consideration for 914:

A power divider is often constructed as a directional coupler, with the isolated port permanently terminated in a matched load (making it effectively a three-port device). There is no essential difference between the two devices. The term

547:

A stub is a short length of line that branches to the side of a main line. The end of the stub is often left open- or short-circuited, but may also be terminated with a lumped component. A stub can be used on its own (for instance, for

1166: 1065:; however, circulators are an exception. There are several equivalent ways to define or represent reciprocity. A convenient one for circuits at microwave frequencies (where distributed-element circuits are used) is in terms of their 732:

the manufacturing tolerances become tighter and are eventually greater than the construction method can achieve. However, after a small number of iterations, the performance is close to that of a true fractal. These may be called

519:

in high-end radio transmitters (marine, military, amateur radio), electronic circuits can be implemented as mechanical components; this is done largely because of the high quality of the mechanical resonators. They are used in the

681:

of wire in a cavity; one end is unconnected, and the other is bonded to the cavity wall. Although they are superficially similar to lumped inductors, helical resonators are distributed-element components and are used in the

389:. The designer does not usually intend to implement the circuit in this form; it is an intermediate step in the design process. Distributed-element designs with conductor pairs are limited to a few specialised uses, such as 797:, the taper is itself the antenna. Horn antennae, like other tapers, are often linear, but the best match is obtained with an exponential curve. The Vivaldi antenna is a flat (slot) version of the exponential taper. 208:, respectively. The distributed-element model is used when this assumption no longer holds, and these properties are considered to be distributed in space. The assumption breaks down when there is significant time for 1315:

The wartime work was mostly unpublished until after the war for security reasons, which made it difficult to ascertain who was responsible for each development. An important centre for this research was the

764:. The first three are closed curves, suitable for patch antennae. The latter two are open curves with terminations on opposite sides of the fractal. This makes them suitable for use where a connection in 376:

Several types of transmission line exist, and any of them can be used to construct distributed-element circuits. The oldest (and still most widely used) is a pair of conductors; its most common form is

1320:(Rad Lab), but work was also done elsewhere in the US and Britain. The Rad Lab work was published by Fano and Lawson. Another wartime development was the hybrid ring. This work was carried out at 453:

The majority of modern distributed-element circuits use planar transmission lines, especially those in mass-produced consumer items. There are several forms of planar line, but the kind known as

911:. For power flowing in the reverse direction and entering the output port, a reciprocal situation occurs; some power is coupled to the isolated port, but none is coupled to the coupled port. 927:

Distributed-element designs rely on an element length of one-quarter wavelength (or some other length); this will hold true at only one frequency. Simple designs, therefore, have a limited

2902:

Janković, Nikolina; Zemlyakov, Kiril; Geschke, Riana Helena; Vendik, Irina; Crnojević-Bengin, Vesna, "Fractal-based multi-band microstrip filters", ch. 6 in, Crnojević-Bengin, Vesna (ed),

563:

Departures from constructing with uniform transmission lines in distributed-element circuits are rare. One such departure that is widely used is the radial stub, which is shaped like a

321:

An important difference between distributed-element circuits and lumped-element circuits is that the frequency response of a distributed circuit periodically repeats as shown in the

1307:

purposes. These frequencies require long antennae for efficient operation, and this led to work on higher-frequency systems. A key breakthrough was the 1940 introduction of the

552:), or several of them can be used together in a more complex circuit such as a filter. A stub can be designed as the equivalent of a lumped capacitor, inductor, or resonator. 777:

reduces the mismatch effects that a direct join would cause. If the change in cross-section is not too great, no other matching circuitry may be needed. Tapers can provide

363:. Distributed-element designs tend to have greater power-handling capability; with a lumped component, all the energy passed by a circuit is concentrated in a small volume. 502:

are possible. These sometimes exist simultaneously, and this situation has no analogy in conducting lines. Waveguides have the advantages of lower loss and higher quality

931:

over which they will work successfully. Like impedance matching networks, a wide-band design requires multiple sections and the design begins to resemble a filter.

1324:, and was published after the war by W. A. Tyrrell. Tyrrell describes hybrid rings implemented in waveguide, and analyses them in terms of the well-known waveguide 310:

consisting of capacitors and inductors can be directly converted into a distributed circuit with a one-to-one correspondence between the elements of each circuit.

1079: 640:

is an empty (or sometimes dielectric-filled) space surrounded by conducting walls. Apertures in the walls couple the resonator to the rest of the circuit.

970:. Each of its four ports is connected to a ring of transmission line at a different point. Waves travel in opposite directions around the ring, setting up 274:

integrated circuits. This choice is particularly significant for hand-held devices, because lumped-element designs generally result in a smaller product.

498:

Many distributed-element designs can be directly implemented in waveguide. However, there is an additional complication with waveguides in that multiple

962:(a lumped device used in telephones), it now has a broader meaning. A widely used distributed-element hybrid which does not use coupled lines is the 525:

mechanical circuits. This is because the speed at which signals travel through mechanical media is much lower than the speed of electrical signals.

461:

and hence is cheap to make. It also lends itself to integration with lumped circuits on the same board. Other forms of printed planar lines include

242:

There is no clear-cut demarcation in the frequency at which these models should be used. Although the changeover is usually somewhere in the 100-to-

1213: 889: 583: 470: 355: 122: 3261:

Ramadan, Ali; Al-Husseini, Mohammed; Kabalan Karim Y; El-Hajj, Ali, "Fractal-shaped reconfigurable antennas", ch. 10 in, Nasimuddin, Nasimuddin,

648:. Cavity resonators can be used in many media, but are most naturally formed in waveguide from the already existing metal walls of the guide. 345:). There is no equivalent in lumped circuits for a fixed delay, although an approximation could be constructed for a limited frequency range. 3437: 978:

results in a null; no power will leave a port set at that point. At other points, constructive interference maximises the power transferred.

1058:

is used to reflect back more power than it received. The circulator is used to direct the input and output power flows to separate ports.

431:

coaxial line is used for low-loss and high-power applications. Distributed-element circuits in other media still commonly transition to

148:

in the field soon led to broader applications. They can now be found in domestic products such as satellite dishes and mobile phones.

616: 623:; in this role, it is called an impedance inverter. This structure can be used in filters to implement a lumped-element prototype in 533:

There are several structures that are repeatedly used in distributed-element circuits. Some of the common ones are described below.

567:. They are often used in pairs, one on either side of the main transmission line. Such pairs are called butterfly or bowtie stubs. 872:; the structure can become quite complex. For simple, narrow-band requirements, a single resonator may suffice (such as a stub or 3258:, interview no. 005 for the IEEE History Centre, 3 March 1973, Engineering and Technology History Wiki, retrieved 15 April 2018. 212:

to travel from one terminal of a component to the other; "significant", in this context, implies enough time for a noticeable

3419: 3404: 3389: 3363: 3348: 3322: 3307: 3270: 3248: 3222: 3207: 3192: 3177: 3162: 3068: 3053: 3038: 3023: 3008: 2982: 2967: 2952: 2926: 2911: 2896: 2881: 2866: 2851: 2836: 2821: 2806: 2759: 2744: 2718: 2703: 2673: 2658: 2632: 2617: 2602: 2587: 2572: 2546: 2531: 2502: 2487: 2472: 2457: 2442: 2416: 805:

Resistive elements are generally not useful in a distributed-element circuit. However, distributed resistors may be used in

840: 105:

A phenomenon commonly used in distributed-element circuits is that a length of transmission line can be made to behave as a

1374:

which overcame some practical limitations of Richards theory, published by Kuroda in 1955. According to Nathan Cohen, the

3447: 1270: 928: 263: 246:

range, the technological scale is also significant; miniaturised circuits can use the lumped model at a higher frequency.

1196:

Distributed elements are usually passive, but most applications will require active components in some role. A microwave

1062: 813:. In planar media they can be implemented as a meandering line of high-resistance material, or as a deposited patch of 482: 87: 1332: 3103:"The use of coaxial and balanced transmission lines in filters and wide band transformers for high radio frequencies" 3442: 1278: 1244: 835: 118: 3239:

Penn, Stuart; Alford, Neil, "Ceramic dielectrics for microwave applications", ch. 10 in, Nalwa, Hari Singh (ed),

1371: 1208:, and some passive components) are discrete. The active components may be packaged, or they may be placed on the 1197: 303: 259: 173: 50:

or other distributed components. These circuits perform the same functions as conventional circuits composed of

24: 1363: 1317: 1252: 1185: 991: 806: 778: 729: 448: 299: 255: 894: 1209: 987: 555: 282: 251: 95: 51: 3102: 1367: 810: 315: 919:

is usually used when the coupling factor (the proportion of power reaching the coupled port) is low, and

1219: 1039:

to protect a transmitter (or other equipment) from damage due to reflections from the antenna, and as a

458: 436: 247: 185: 160: 132:

Distributed-element circuits were studied during the 1920s and 1930s but did not become important until

75: 1348: 2666:

A History of Engineering and Science in the Bell System: Volume 5: Communications Sciences (1925–1980)

1375: 1227: 1047: 1036: 862: 657: 603: 209: 177: 79: 821:

material. In waveguide, a card of microwave absorbent material can be inserted into the waveguide.

1274: 1051: 753: 417: 342: 318:

to determine its behaviour, finding useful structures is a matter of trial and error or guesswork.

306:, where all the elements are the same length. With commensurate circuits, a lumped circuit design 1231: 959: 880:

difference from a standard filter design is that the filter's source and load impedances differ.

765: 576: 549: 542: 338: 267: 181: 114: 110: 43: 3132: 907:. None of the power entering the input port is coupled to the fourth port, usually known as the 151: 1351:. Although stripline was another wartime invention, its details were not published until 1951. 1277:, an effect which was not well understood at the time. Heaviside's analysis, now known as the 3415: 3400: 3385: 3359: 3344: 3318: 3303: 3266: 3244: 3218: 3203: 3188: 3173: 3158: 3143: 3128: 3064: 3049: 3034: 3019: 3004: 2978: 2963: 2948: 2922: 2907: 2892: 2877: 2862: 2847: 2832: 2817: 2802: 2787: 2770: 2755: 2740: 2714: 2699: 2684: 2669: 2654: 2628: 2613: 2598: 2583: 2568: 2542: 2527: 2498: 2483: 2468: 2453: 2438: 2412: 1384: 975: 721: 672: 644:

occurs due to electromagnetic waves reflected back and forth from the cavity walls setting up

516: 432: 334: 330: 326: 295: 145: 47: 1269:

in 1881. Heaviside used it to find a correct description of the behaviour of signals on the

789:

Tapers can be used to match a transmission line to an antenna. In some designs, such as the

270:

can use lumped designs at higher frequencies than printed circuits, and this is done in some

16:

Electrical circuits composed of lengths of transmission lines or other distributed components

1359: 1308: 1266: 1248: 1070: 1032: 967: 939: 869: 783: 749: 637: 492: 322: 307: 287: 1180: 898:

Microstrip sawtooth directional coupler, a variant of the coupled-lines directional coupler

1288: 1161:{\displaystyle ={\begin{pmatrix}0&0&1\\1&0&0\\0&1&0\end{pmatrix}}} 873: 868:

As with lumped-element filters, the more elements used, the closer the filter comes to an

794: 699: 624: 564: 521: 398: 271: 156: 2593:

Cohen, Nathan, "Fractal antenna and fractal resonator primer", ch. 8 in, Frame, Michael,

1212:

in chip form without individual packaging to reduce size and eliminate packaging-induced

947:

A directional coupler which splits power equally between the output and coupled ports (a

31:. The distributed-element circuitry is centre and left of centre, and is constructed in 983: 745: 620: 499: 386: 382: 99: 70:

frequencies, where conventional components are difficult (or impossible) to implement.

28: 1200:

uses distributed elements for many passive components, but active components (such as

487: 3431: 1379: 999: 995: 971: 757: 645: 575:

Coupled lines are two transmission lines between which there is some electromagnetic

424: 294:

The overwhelming majority of distributed-element circuits are composed of lengths of

213: 83: 598: 2827:

Hilty, Kurt, "Attenuation measurement", pp. 422–439 in, Dyer, Stephen A (ed),

2679:

Fano, R M; Lawson, A W, "Design of microwave filters", ch. 10 in, Ragan, G L (ed),

1336: 1312:

extending the filter technology from the coaxial domain into the waveguide domain.

1304: 1300: 1282: 1273:. Transmission of early transatlantic telegraph had been difficult and slow due to 1073:. From the definition of a circulator, it is clear that this will not be the case, 1066: 1002: 790: 378: 141: 133: 2958:

Lacomme, Philippe; Marchais, Jean-Claude; Hardange, Jean-Philippe; Normant, Eric,

3014:

Magnusson, Philip C; Weisshaar, Andreas; Tripathi, Vijai K; Alexander, Gerald C,

3329: 2423: 1013: 761: 725: 704: 390: 189: 63: 3255: 2554:"The introduction of the loading coil: George A. Campbell and Michael I. Pupin" 1265:

Distributed-element modelling was first used in electrical network analysis by

412: 354:

miniaturise components rather than switching to distributed elements. However,

163:(left), and as a distributed-element design printed on the board itself (right) 3123:

Matthaei, G L, "Comb-line band-pass filters of narrow or moderate bandwidth",

3094: 3085: 3076: 2639: 1352: 1205: 1055: 1025: 818: 619:. This has the useful property of transforming any impedance network into its 454: 428: 221: 193: 126: 32: 3370: 3334:

Transactions of the IRE Professional Group on Microwave Theory and Techniques

3277: 3241:

Handbook of Low and High Dielectric Constant Materials and Their Applications

2725: 628:

of two cascaded transmission lines with different characteristic impedances.

3113: 2989: 1344: 1325: 1321: 1293: 1257: 845: 814: 717: 641: 587: 503: 462: 401: 394: 232: 217: 201: 106: 67: 55: 3229: 337:; distributed forms are an irrational function. Another difference is that 3147: 2933: 586:. Another property of coupled lines is that they act as a pair of coupled 1296:

radio department asked him to assist with coaxial and waveguide filters.

1040: 849: 607: 360: 325:

example; the equivalent lumped circuit does not. This is a result of the

205: 197: 59: 3200:

A Practical Design of Lumped, Semi-lumped & Microwave Cavity Filters

2791: 2781: 2774: 2688: 2625:

Filter Design for Satellite Communications: Helical Resonator Technology

341:

lengths of line introduce a fixed delay at all frequencies (assuming an

19: 3140:

Microwave Filters, Impedance-Matching Networks, and Coupling Structures

2553: 1189: 950: 858: 728:

rejection. In practice, a true fractal cannot be made because at each

713: 466: 2726:"Microstrip—a new transmission technique for the kilomegacycle range" 3288: 2780:

Heaviside, Oliver, "Electromagnetic induction and its propagation",

1328:. Other researchers soon published coaxial versions of this device. 1043:

connecting the antenna, transmitter and receiver of a radio system.

117:, and cascaded lines. Circuits built from these components include 2524:

Stripline-like Transmission Lines for Microwave Integrated Circuits

1069:. A reciprocal circuit will have an S-parameter matrix, , which is 1256: 1201: 1179: 1012: 938: 893: 839: 703: 678: 597: 554: 486: 457:

is the most common. It can be manufactured by the same process as

411: 281: 150: 137: 91: 18: 2990:"A History of microwave filter research, design, and development" 2580:

Microwave Electronics: Measurement and Materials Characterization

1061:

Passive circuits, both lumped and distributed, are nearly always

982:

couplers and power dividers, directional couplers can be used in

744:

Fractals that have been used as a circuit component include the

290:

constructed from lumped (top) and distributed components (bottom)

2945:

Distributed Power Amplifiers for RF and Microwave Communications

2874:

Data and Computer Communications: Networking and Internetworking

2578:

Chen, L F; Ong, C K; Neo, C P; Varadan, V V; Varadan, Vijay K,

3382:

Microwave Circuit Design Using Linear and Nonlinear Techniques

2623:

Doumanis, Efstratios; Goussetis, George; Kosmopoulos, Savvas,

2409:

Asymmetric Passive Components in Microwave Integrated Circuits

1223: 687: 683: 227: 216:

change. The amount of phase change is dependent on the wave's

140:. After the war their use was limited to military, space, and 2508:

Barrett, R M, "Etched sheets serve as microwave components",

590:. This property is used in many distributed-element filters. 1387:

in 1987, but he could not get a paper published until 1995.

615:

A single, cascaded line one-quarter wavelength long forms a

420:. One has the cover removed, showing its internal structure. 3031:

Microwave Resonators and Filters for Wireless Communication

3061:

Integrated Microwave Front-ends with Avionics Applications

1285:. It remains the standard analysis of transmission lines. 1184:

Microstrip circuit with discrete transistors in miniature

861:

filters. Many filters are constructed in conjunction with

741:

where it is necessary to distinguish from a true fractal.

2515:

Barrett, R M; Barnes, M H, "Microwave printed circuits",

27:

with distributed elements. The circuitry on the right is

2889:

Handbook of Microwave Technology: Components and devices

109:. Distributed-element components which do this include 3380:

Vendelin, George D; Pavio, Anthony M; Rohde, Ulrich L,

2640:"Broadband logarithmically periodic antenna structures" 2465:

Control Components Using Si, GaAs, and GaN Technologies

943:

Hybrid ring, used to produce sum and difference signals

708:

Three-iteration Hilbert fractal resonator in microstrip

3377:, vol. 35, iss. 11, pp. 1294–1306, November 1947. 2732:, vol. 40, iss. 12, pp. 1644–1650, December 1952. 2450:

Fundamentals of RF and Microwave Transistor Amplifiers

1343:

Planar formats began to be used with the invention of

1100: 469:

and many variations. Planar lines can also be used in

196:

are assumed to be "lumped" at one point in space in a

1188:

packages, capacitors and resistors in chip form, and

1082: 3185:

Radio-Frequency and Microwave Communication Circuits

2932:

Johnson, Robert A; Börner, Manfred; Konno, Masashi,

2769:, vol. 1, pp. 139–140, Copley Publishers, 1925 507:

which militates against its use on the lower bands.

3120:, vol. 10, iss. 6, pp. 479–491, November 1962. 3118:

IRE Transactions on Microwave Theory and Techniques

2428:

IRE Transactions on Microwave Theory and Techniques

1222:consist of a number of amplifying devices (usually 172:Distributed-element circuits are designed with the 159:as conventional discrete components connected on a 3092:Mason, Warren P, "Electromechanical wave filter", 2994:IEEE Transactions: Microwave Theory and Techniques 1160: 884:Power dividers, combiners and directional couplers 582:Coupled lines are a common method of constructing 2799:Ridge Waveguides and Passive Microwave Components 2430:, vol. 6, iss. 4, pp. 369–373, October 1958. 2424:"An analysis of a broad-band coaxial hybrid ring" 3341:Basic Microwave Techniques and Laboratory Manual 3336:, vol. 1, iss. 2, pp. 17–23, November 1953. 2844:Microstrip Filters for RF/Microwave Applications 3295:, vol. 10, iss. 6, pp. 391–397, June 1939. 3089:, filed 25 November 1941, issued 28 March 1944. 2940:, vol. 18, iss. 3, pp. 155–170, July 1971. 2829:Wiley Survey of Instrumentation and Measurement 2560:, vol. 11, no. 1, pp. 36–57, January 1970. 1046:An unusual application of a circulator is in a 3315:Fundamental of Microwave and Radar Engineering 3236:, vol. 5, iss. 2, pp. 104–109, June 1958. 3215:Radio-Frequency Integrated-Circuit Engineering 3202:, Springer Science & Business Media, 2012 3138:Matthaei, George L; Young, Leo; Jones, E M T, 3083:Mason, Warren P, "Wave transmission network", 3080:, filed 25 June 1927, issued 11 November 1930. 2565:Microwave Ring Circuits and Related Structures 2437:, Springer Science & Business Media, 2013 515:In a few specialist applications, such as the 473:, where they are integral to the device chip. 3098:, filed 20 August 1958, issued 25 April 1961. 1035:materials. Uses of circulators include as an 1031:distributed-element circulators are based on 958:. Although "hybrid" originally referred to a 8: 3230:"Synthesis of a class of strip-line filters" 3046:Passive RF and Microwave Integrated Circuits 2996:, pp. 1055–1067, vol. 32, iss. 9, 1984. 2694:Garg, Ramesh; Bahl, Inder; Bozzi, Maurizio, 2595:Benoit Mandelbrot: A Life In Many Dimensions 1358:Important theoretical developments included 724:designs, good in-band performance, and good 2938:IEEE Transactions on Sonics and Ultrasonics 2814:Microwave Mixer Technology and Applications 3284:, vol. 36, iss. 2, pp. 217–220, 1948. 1017:A coaxial ferrite circulator operating at 974:. At some points on the ring, destructive 3412:Passive Microwave Components and Antennas 2934:"Mechanical filters—a review of progress" 2904:Advances in Multi-Band Microstrip Filters 1245:Distributed-element filter § History 1095: 1081: 254:are larger than equivalent designs using 2921:, John Wiley & Sons Australia, 1983 2651:Foundations of Microstrip Circuit Design 2644:1958 IRE International Convention Record 1331:George Matthaei led a research group at 471:monolithic microwave integrated circuits 3016:Transmission Lines and Wave Propagation 2752:The Cable Television Technical Handbook 2512:, vol. 25, pp. 114–118, June 1952. 1395: 1281:, identified the problem and suggested 1253:Planar transmission line § History 890:Power dividers and directional couplers 584:power dividers and directional couplers 262:are smaller than PCB technologies, and 3127:, vol. 6, pp. 82–91, August 1963 2943:Kumar, Narendra; Grebennikov, Andrei, 2859:Theory and Design of Microwave Filters 231:, and are the technology of choice at 3278:"Resistor-transmission-line circuits" 848:hairpin filter (left), followed by a 302:. A further simplification occurs in 7: 3397:The Resource Handbook of Electronics 2649:Edwards, Terry C; Steer, Michael B, 2120:Kumar & Grebennikov, pp. 153–154 1588:Edwards & Steer, pp. 78, 345–347 286:Frequency response of a fifth-order 278:Construction with transmission lines 144:infrastructure, but improvements in 123:power dividers, directional couplers 2977:, Cambridge University Press, 2004 2906:, Cambridge University Press, 2015 2713:, Cambridge University Press, 2017 2668:, AT&T Bell Laboratories, 1984 2646:, New York, 1957, pp. 119–128. 1366:, which was published in 1948, and 3356:CRC Handbook of Electrical Filters 3234:IRE Transactions on Circuit Theory 3109:, vol. 16, pp. 275–302, 1937. 2872:Hura, Gurdeep S; Singhal, Mukesh, 2842:Hong, Jia-Shen G; Lancaster, M J, 2495:Automated Electronic Filter Design 1930:Hong & Lancaster, pp. 109, 235 1505:Henderson & Camargo, pp. 24–25 1378:, invented by Raymond DuHamel and 617:quarter-wave impedance transformer 14: 2919:Mechanical Filters in Electronics 2812:Henderson, Bert; Camargo, Edmar, 2709:Ghione, Giovanni; Pirola, Marco, 2610:Electronics via Waveform Analysis 2539:Radar Imaging of Airborne Targets 2350:Makimoto & Yamashita, pp. 1–2 1644:Bhat & Koul, pp. 10, 602, 622 610:) with stepped impedance matching 90:formats for applications such as 3371:"Hybrid circuits for microwaves" 3339:Sisodia, M L; Raghuvanshi, G S, 3114:"Interdigital band-pass filters" 3074:Mason, Warren P, "Wave filter", 2960:Air and Spaceborne Radar Systems 2739:, Krishna Prakashan Media, 2010 2522:Bhat, Bharathi; Koul, Shiban K, 2026:Ghione & Pirola, pp. 172–173 1603:Edwards & Steer, pp. 347–348 3256:"Oral-History: Warren P. Mason" 2681:Microwave Transmission Circuits 2482:, Technical Publications, 2009 1981:Sisodia & Raghuvansh, p. 70 1489:Hura & Singhal, pp. 178–179 1249:Waveguide filter § History 1192:filters as distributed elements 78:for consumer products, such as 3414:, BoD – Books on Demand, 2010 3384:, John Wiley & Sons, 2005 3354:Taylor, John; Huang, Qiuting, 3343:, New Age International, 1987 3265:, BoD – Books on Demand, 2011 3217:, John Wiley & Sons, 2015 3187:, John Wiley & Sons, 2004 3172:, John Wiley & Sons, 2005 2846:, John Wiley & Sons, 2004 2831:, John Wiley & Sons, 2004 2786:, pp. 79–81, 3 June 1887 2696:Microstrip Lines and Slotlines 2653:, John Wiley & Sons, 2016 2582:, John Wiley & Sons, 2004 2567:, John Wiley & Sons, 2004 2526:, New Age International, 1989 2452:, John Wiley & Sons, 2009 2411:, John Wiley & Sons, 2006 2386:Levy & Cohn, pp. 1056–1057 2111:Bhat & Khoul, pp. 9–10, 15 2102:Maloratsky (2004), pp. 285–286 2017:Chang & Hsieh, pp. 197–198 1933:Makimoto & Yamashita, p. 2 1680:Penn & Alford, pp. 524–530 1629:Chang & Hsieh, pp. 227–229 1560:Taylor & Huang pp. 353–358 1539:Ghione & Pirola, pp. 18–19 1089: 1083: 924:high-coupling power dividers. 830:Filters and impedance matching 439:for interconnection purposes. 264:monolithic integrated circuits 1: 3107:Bell System Technical Journal 2737:Electro Magnetic Field Theory 2563:Chang, Kai; Hsieh, Lung-Hwa, 2519:, vol. 46, 16 September 1951. 2250:Sheingold & Morita (1953) 1999:Bhat & Khoul, pp. 622–627 1271:transatlantic telegraph cable 3438:Distributed element circuits 3317:, S. Chand Publishing, 2011 3300:Microwave Electronic Devices 2975:Planar Microwave Engineering 2724:Grieg, D D; Englemann, H F, 2480:Antenna And Wave Propagation 2422:Albanese, V J; Peyser, W P, 2305:Levy and Cohn, pp. 1057–1059 2253:Albanese & Peyser (1958) 1632:Bhat & Koul, pp. 602–609 483:Waveguide (electromagnetism) 349:Advantages and disadvantages 220:(and inversely dependent on 40:Distributed-element circuits 3328:Sheingold, L S; Morita, T, 3029:Makimoto, M; Yamashita, S, 3001:Microwave Circulator Design 2314:Barrett & Barnes (1951) 2184:Fagen & Millman, p. 108 1834:Edwards & Steer, p. 493 1333:Stanford Research Institute 66:. They are used mostly at 3464: 3293:Journal of Applied Physics 3198:Natarajan, Dhanasekharan, 2478:Bakshi, U A; Bakshi, A V, 2371:First English publication: 2323:Grieg and Englemann (1952) 1548:Ghione & Pirola, p. 18 1242: 1023: 887: 836:Distributed-element filter 833: 697: 670: 655: 540: 480: 446: 304:commensurate line circuits 260:Hybrid integrated circuits 2597:, World Scientific, 2015 2041:Maloratsky (2004), p. 117 2038:Chang & Hsieh, p. 227 2008:Maloratsky (2004), p. 117 1972:Maloratsky (2004), p. 160 1283:methods for overcoming it 1198:hybrid integrated circuit 677:A helical resonator is a 174:distributed-element model 136:, when they were used in 25:low-noise block converter 3101:Mason, W P; Sykes, R A, 2664:Fagen, M D; Millman, S, 2638:DuHamell, R; Isbell, D, 2435:Microwave Systems Design 2374:Ozaki & Ishii (1958) 2229:Levy & Cohn, p. 1055 2220:Fano & Lawson (1948) 2211:Levy & Cohn, p. 1055 2187:Levy & Cohn, p. 1055 1891:Maloratsky (2012), p. 69 1364:commensurate line theory 1318:MIT Radiation Laboratory 988:frequency discriminators 449:Planar transmission line 416:A collection of coaxial 329:of lumped forms being a 300:characteristic impedance 256:surface-mount technology 176:, an alternative to the 82:. They are also made in 3289:"Dielectric resonators" 3243:, Academic Press, 1999 2962:, William Andrew, 2001 2891:, Academic Press, 1995 2138:Heaviside (1887), p. 81 1279:telegrapher's equations 266:are smaller than both. 252:through-hole technology 96:satellite communication 46:composed of lengths of 3375:Proceedings of the IRE 3282:Proceedings of the IRE 3254:Polkinghorn, Frank A, 2730:Proceedings of the IRE 2558:Technology and Culture 1563:Johnson (1983), p. 102 1262: 1220:Distributed amplifiers 1193: 1162: 1021: 944: 899: 853: 801:Distributed resistance 709: 611: 560: 495: 459:printed circuit boards 421: 291: 248:Printed circuit boards 164: 36: 3170:Modern Antenna Design 3157:, Artech House, 2008 3095:U.S. patent 2,981,905 3086:U.S. patent 2,345,491 3077:U.S. patent 2,345,491 3063:, Artech House, 2012 3003:, Artech House, 2014 2999:Linkhart, Douglas K, 2947:, Artech House, 2015 2816:, Artech House, 2013 2754:, Artech House, 1985 2711:Microwave Electronics 2698:, Artech House, 2013 2627:, Artech House, 2015 2467:, Artech House, 2014 2332:Bhat & Koul, p. 3 1260: 1183: 1163: 1016: 954:coupler) is called a 942: 897: 863:dielectric resonators 843: 739:finite-order fractals 707: 601: 559:Butterfly stub filter 558: 490: 415: 285: 210:electromagnetic waves 186:electrical resistance 180:in which the passive 161:printed circuit board 154: 76:printed circuit board 22: 3448:Microwave technology 3410:Zhurbenko, Vitaliy, 3228:Ozaki, H; Ishii, J, 3168:Milligan, Thomas A, 3155:Modern Radar Systems 2988:Levy, R; Cohn, S B, 2683:, McGraw-Hill, 1948 1573:(1971), pp. 155, 169 1527:Natarajan, pp. 11–12 1376:log-periodic antenna 1228:multistage amplifier 1172:the device to work. 1080: 1048:reflection amplifier 658:Dielectric resonator 652:Dielectric resonator 604:orthomode transducer 418:directional couplers 178:lumped-element model 80:satellite television 54:components, such as 3395:Whitaker, Jerry C, 3330:"A coaxial magic-T" 3263:Microstrip Antennas 3183:Misra, Devendra K, 3059:Maloratsky, Leo G, 3044:Maloratsky, Leo G, 2917:Johnson, Robert A, 2765:Heaviside, Oliver, 2552:Brittain, James E, 2056:Sharma, pp. 175–176 1963:Bahl (2009), p. 149 1897:Bahl (2014), p. 214 1870:Bakshi & Bakshi 1671:Hunter, pp. 209–210 1455:Hunter, pp. 139–140 1437:Hunter, pp. 137–138 1368:Kuroda's identities 1052:negative resistance 917:directional coupler 316:Maxwell's equations 268:Integrated circuits 182:electrical elements 44:electrical circuits 3399:, CRC Press, 2000 3358:, CRC Press, 1997 3276:Richards, Paul I, 3018:, CRC Press, 2000 2876:, CRC Press, 2001 2541:, CRC Press, 1999 2395:Cohen, pp. 210–211 2190:Polkinghorn (1973) 1909:Hilty, pp. 426–427 1831:, pp. 404–406, 540 1502:Craig, pp. 291–292 1263: 1194: 1158: 1152: 1022: 960:hybrid transformer 945: 900: 854: 710: 612: 565:sector of a circle 561: 550:impedance matching 543:Stub (electronics) 529:Circuit components 517:mechanical filters 496: 433:coaxial connectors 422: 356:parasitic elements 292: 235:frequencies above 165: 48:transmission lines 37: 3443:Radio electronics 3302:, Springer, 2012 3287:Richtmeyer, R D, 3142:McGraw-Hill 1964 3125:Microwave Journal 3048:, Elsevier, 2004 3033:, Springer, 2013 2767:Electrical Papers 2612:, Springer, 2012 2497:, Springer, 2016 2341:Richtmeyer (1939) 2093:Roer, pp. 255–256 1849:Zhurbenko, p. 311 1802:Zhurbenko, p. 310 1474:Nguyen, pp. 27–28 1385:Benoit Mandelbrot 1349:Robert M. Barrett 1176:Active components 730:fractal iteration 673:Helical resonator 667:Helical resonator 372:Paired conductors 339:cascade-connected 335:complex frequency 331:rational function 327:transfer function 296:transmission line 168:Circuit modelling 146:materials science 3455: 3153:Meikle, Hamish, 3097: 3088: 3079: 2887:Ishii, T Koryu, 2608:Craig, Edwin C, 2493:Banerjee, Amal, 2396: 2393: 2387: 2384: 2378: 2366: 2360: 2357: 2351: 2348: 2342: 2339: 2333: 2330: 2324: 2321: 2315: 2312: 2306: 2303: 2297: 2290: 2284: 2281: 2275: 2272: 2266: 2263: 2257: 2245: 2239: 2236: 2230: 2227: 2221: 2218: 2212: 2209: 2203: 2200: 2194: 2172: 2166: 2163: 2157: 2154: 2148: 2145: 2139: 2136: 2130: 2129:Heaviside (1925) 2127: 2121: 2118: 2112: 2109: 2103: 2100: 2094: 2091: 2085: 2069: 2063: 2051: 2045: 2033: 2027: 2024: 2018: 2015: 2009: 2006: 2000: 1997: 1991: 1988: 1982: 1979: 1973: 1970: 1964: 1961: 1955: 1952: 1946: 1943: 1937: 1925: 1919: 1916: 1910: 1907: 1901: 1886: 1880: 1876:Milligan, p. 513 1865: 1859: 1844: 1838: 1822: 1816: 1809: 1803: 1800: 1794: 1787: 1781: 1774: 1768: 1761: 1755: 1748: 1742: 1735: 1729: 1722: 1716: 1709: 1703: 1692:Whitaker, p. 227 1687: 1681: 1678: 1672: 1669: 1663: 1662:Helszajn, p. 189 1660: 1654: 1651: 1645: 1642: 1636: 1610: 1604: 1601: 1595: 1583: 1577: 1555: 1549: 1546: 1540: 1537: 1531: 1522: 1516: 1484: 1478: 1462: 1456: 1453: 1447: 1444: 1438: 1435: 1429: 1413: 1407: 1400: 1383:by a lecture of 1360:Paul I. Richards 1309:cavity magnetron 1267:Oliver Heaviside 1261:Oliver Heaviside 1167: 1165: 1164: 1159: 1157: 1156: 1020: 968:rat-race coupler 953: 784:Chebychev filter 754:Sierpiński curve 750:Minkowski island 638:cavity resonator 632:Cavity resonator 493:waveguide filter 323:Chebyshev filter 288:Chebyshev filter 245: 238: 230: 3463: 3462: 3458: 3457: 3456: 3454: 3453: 3452: 3428: 3427: 3112:Matthaei, G L, 3093: 3084: 3075: 2973:Lee, Thomas H, 2783:The Electrician 2750:Harrel, Bobby, 2537:Borden, Brett, 2463:Bahl, Inder J, 2448:Bahl, Inder J, 2404: 2399: 2394: 2390: 2385: 2381: 2377: 2367: 2363: 2359:Richards (1948) 2358: 2354: 2349: 2345: 2340: 2336: 2331: 2327: 2322: 2318: 2313: 2309: 2304: 2300: 2291: 2287: 2283:Matthaei (1963) 2282: 2278: 2274:Matthaei (1962) 2273: 2269: 2264: 2260: 2256: 2246: 2242: 2237: 2233: 2228: 2224: 2219: 2215: 2210: 2206: 2201: 2197: 2193: 2173: 2169: 2164: 2160: 2155: 2151: 2147:Brittain, p. 39 2146: 2142: 2137: 2133: 2128: 2124: 2119: 2115: 2110: 2106: 2101: 2097: 2092: 2088: 2084: 2070: 2066: 2062: 2059:Linkhart, p. 29 2052: 2048: 2044: 2034: 2030: 2025: 2021: 2016: 2012: 2007: 2003: 1998: 1994: 1989: 1985: 1980: 1976: 1971: 1967: 1962: 1958: 1953: 1949: 1945:Harrell, p. 150 1944: 1940: 1936: 1926: 1922: 1917: 1913: 1908: 1904: 1900: 1887: 1883: 1879: 1866: 1862: 1858: 1845: 1841: 1837: 1823: 1819: 1810: 1806: 1801: 1797: 1788: 1784: 1775: 1771: 1762: 1758: 1749: 1745: 1736: 1732: 1723: 1719: 1710: 1706: 1702: 1688: 1684: 1679: 1675: 1670: 1666: 1661: 1657: 1652: 1648: 1643: 1639: 1635: 1611: 1607: 1602: 1598: 1594: 1591:Banerjee, p. 74 1584: 1580: 1576: 1556: 1552: 1547: 1543: 1538: 1534: 1530: 1523: 1519: 1515: 1485: 1481: 1477: 1463: 1459: 1454: 1450: 1445: 1441: 1436: 1432: 1428: 1414: 1410: 1401: 1397: 1393: 1335:which included 1289:Warren P. Mason 1255: 1241: 1178: 1151: 1150: 1145: 1140: 1134: 1133: 1128: 1123: 1117: 1116: 1111: 1106: 1096: 1078: 1077: 1028: 1018: 1011: 984:balanced mixers 948: 937: 892: 886: 874:spurline filter 838: 832: 827: 803: 795:Vivaldi antenna 774: 702: 700:Fractal antenna 696: 675: 669: 660: 654: 634: 625:ladder topology 596: 573: 545: 539: 531: 522:radio frequency 513: 485: 479: 451: 445: 435:at the circuit 410: 387:telegraph poles 383:telephone lines 374: 369: 351: 280: 272:radio frequency 243: 236: 225: 170: 157:low-pass filter 100:microwave links 29:lumped elements 17: 12: 11: 5: 3461: 3459: 3451: 3450: 3445: 3440: 3430: 3429: 3424: 3423: 3408: 3393: 3378: 3369:Tyrrell, W A, 3367: 3352: 3337: 3326: 3311: 3296: 3285: 3274: 3259: 3252: 3237: 3226: 3211: 3196: 3181: 3166: 3151: 3136: 3121: 3110: 3099: 3090: 3081: 3072: 3057: 3042: 3027: 3012: 2997: 2986: 2971: 2956: 2941: 2930: 2915: 2900: 2885: 2870: 2855: 2840: 2825: 2810: 2795: 2778: 2763: 2748: 2733: 2722: 2707: 2692: 2677: 2662: 2647: 2636: 2621: 2606: 2591: 2576: 2561: 2550: 2535: 2520: 2513: 2506: 2491: 2476: 2461: 2446: 2433:Awang, Zaiki, 2431: 2420: 2407:Ahn, Hee-Ran, 2403: 2400: 2398: 2397: 2388: 2379: 2376: 2375: 2372: 2368: 2361: 2352: 2343: 2334: 2325: 2316: 2307: 2298: 2285: 2276: 2267: 2258: 2255: 2254: 2251: 2247: 2240: 2238:Tyrrell (1947) 2231: 2222: 2213: 2204: 2195: 2192: 2191: 2188: 2185: 2182: 2181:(1971), p. 155 2174: 2167: 2158: 2149: 2140: 2131: 2122: 2113: 2104: 2095: 2086: 2083: 2082: 2075: 2071: 2064: 2061: 2060: 2057: 2053: 2046: 2043: 2042: 2039: 2035: 2028: 2019: 2010: 2001: 1992: 1983: 1974: 1965: 1956: 1947: 1938: 1935: 1934: 1931: 1927: 1920: 1911: 1902: 1899: 1898: 1895: 1892: 1888: 1881: 1878: 1877: 1874: 1871: 1867: 1860: 1857: 1856: 1853: 1850: 1846: 1839: 1836: 1835: 1832: 1824: 1817: 1804: 1795: 1782: 1769: 1756: 1743: 1730: 1717: 1704: 1701: 1700: 1693: 1689: 1682: 1673: 1664: 1655: 1646: 1637: 1634: 1633: 1630: 1627: 1620: 1612: 1605: 1596: 1593: 1592: 1589: 1585: 1578: 1575: 1574: 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3460: 3449: 3446: 3444: 3441: 3439: 3436: 3435: 3433: 3426: 3421: 3417: 3413: 3409: 3406: 3402: 3398: 3394: 3391: 3387: 3383: 3379: 3376: 3372: 3368: 3365: 3361: 3357: 3353: 3350: 3346: 3342: 3338: 3335: 3331: 3327: 3324: 3320: 3316: 3313:Sharma, K K, 3312: 3309: 3305: 3301: 3297: 3294: 3290: 3286: 3283: 3279: 3275: 3272: 3268: 3264: 3260: 3257: 3253: 3250: 3246: 3242: 3238: 3235: 3231: 3227: 3224: 3220: 3216: 3213:Nguyen, Cam, 3212: 3209: 3205: 3201: 3197: 3194: 3190: 3186: 3182: 3179: 3175: 3171: 3167: 3164: 3160: 3156: 3152: 3149: 3145: 3141: 3137: 3134: 3130: 3126: 3122: 3119: 3115: 3111: 3108: 3104: 3100: 3096: 3091: 3087: 3082: 3078: 3073: 3070: 3066: 3062: 3058: 3055: 3051: 3047: 3043: 3040: 3036: 3032: 3028: 3025: 3021: 3017: 3013: 3010: 3006: 3002: 2998: 2995: 2991: 2987: 2984: 2980: 2976: 2972: 2969: 2965: 2961: 2957: 2954: 2950: 2946: 2942: 2939: 2935: 2931: 2928: 2924: 2920: 2916: 2913: 2909: 2905: 2901: 2898: 2894: 2890: 2886: 2883: 2879: 2875: 2871: 2868: 2864: 2860: 2857:Hunter, Ian, 2856: 2853: 2849: 2845: 2841: 2838: 2834: 2830: 2826: 2823: 2819: 2815: 2811: 2808: 2804: 2800: 2797:Helszajn, J, 2796: 2793: 2789: 2785: 2784: 2779: 2776: 2772: 2768: 2764: 2761: 2757: 2753: 2749: 2746: 2742: 2738: 2734: 2731: 2727: 2723: 2720: 2716: 2712: 2708: 2705: 2701: 2697: 2693: 2690: 2686: 2682: 2678: 2675: 2671: 2667: 2663: 2660: 2656: 2652: 2648: 2645: 2641: 2637: 2634: 2630: 2626: 2622: 2619: 2615: 2611: 2607: 2604: 2600: 2596: 2592: 2589: 2585: 2581: 2577: 2574: 2570: 2566: 2562: 2559: 2555: 2551: 2548: 2544: 2540: 2536: 2533: 2529: 2525: 2521: 2518: 2517:Radio TV News 2514: 2511: 2507: 2504: 2500: 2496: 2492: 2489: 2485: 2481: 2477: 2474: 2470: 2466: 2462: 2459: 2455: 2451: 2447: 2444: 2440: 2436: 2432: 2429: 2425: 2421: 2418: 2414: 2410: 2406: 2405: 2401: 2392: 2389: 2383: 2380: 2373: 2370: 2369: 2365: 2362: 2356: 2353: 2347: 2344: 2338: 2335: 2329: 2326: 2320: 2317: 2311: 2308: 2302: 2299: 2295: 2289: 2286: 2280: 2277: 2271: 2268: 2262: 2259: 2252: 2249: 2248: 2244: 2241: 2235: 2232: 2226: 2223: 2217: 2214: 2208: 2205: 2199: 2196: 2189: 2186: 2183: 2180: 2176: 2175: 2171: 2168: 2162: 2159: 2153: 2150: 2144: 2141: 2135: 2132: 2126: 2123: 2117: 2114: 2108: 2105: 2099: 2096: 2090: 2087: 2080: 2076: 2074:Meikle, p. 91 2073: 2072: 2068: 2065: 2058: 2055: 2054: 2050: 2047: 2040: 2037: 2036: 2032: 2029: 2023: 2020: 2014: 2011: 2005: 2002: 1996: 1993: 1990:Ishii, p. 226 1987: 1984: 1978: 1975: 1969: 1966: 1960: 1957: 1954:Awang, p. 296 1951: 1948: 1942: 1939: 1932: 1929: 1928: 1924: 1921: 1918:Cohen, p. 220 1915: 1912: 1906: 1903: 1896: 1894:Hilty, p. 425 1893: 1890: 1889: 1885: 1882: 1875: 1873:pp. 3-68–3-70 1872: 1869: 1868: 1864: 1861: 1854: 1852:Misra, p. 276 1851: 1848: 1847: 1843: 1840: 1833: 1830: 1826: 1825: 1821: 1818: 1815:, pp. 180–181 1814: 1808: 1805: 1799: 1796: 1792: 1786: 1783: 1779: 1773: 1770: 1766: 1760: 1757: 1754:, pp. 191–192 1753: 1747: 1744: 1740: 1734: 1731: 1727: 1721: 1718: 1714: 1708: 1705: 1698: 1694: 1691: 1690: 1686: 1683: 1677: 1674: 1668: 1665: 1659: 1656: 1650: 1647: 1641: 1638: 1631: 1628: 1625: 1621: 1618: 1614: 1613: 1609: 1606: 1600: 1597: 1590: 1587: 1586: 1582: 1579: 1572: 1568: 1565: 1562: 1559: 1558: 1554: 1551: 1545: 1542: 1536: 1533: 1526: 1525: 1521: 1518: 1511: 1507: 1504: 1501: 1499:Gupta, p. 5.5 1498: 1495: 1491: 1488: 1487: 1483: 1480: 1473: 1470: 1466: 1465: 1461: 1458: 1452: 1449: 1443: 1440: 1434: 1431: 1424: 1420: 1418:Nguyen, p. 28 1417: 1416: 1412: 1409: 1405: 1399: 1396: 1390: 1388: 1386: 1381: 1380:Dwight Isbell 1377: 1373: 1369: 1365: 1361: 1356: 1354: 1350: 1346: 1341: 1338: 1334: 1329: 1327: 1323: 1319: 1313: 1310: 1306: 1302: 1297: 1295: 1290: 1286: 1284: 1280: 1276: 1272: 1268: 1259: 1254: 1250: 1246: 1238: 1236: 1233: 1229: 1225: 1221: 1217: 1215: 1211: 1207: 1203: 1199: 1191: 1187: 1186:surface-mount 1182: 1175: 1173: 1153: 1147: 1142: 1137: 1130: 1125: 1120: 1113: 1108: 1103: 1097: 1092: 1086: 1076: 1075: 1074: 1072: 1068: 1064: 1059: 1057: 1053: 1049: 1044: 1042: 1038: 1034: 1027: 1015: 1008: 1006: 1004: 1001: 1000:antenna array 997: 993: 989: 985: 979: 977: 973: 969: 965: 961: 957: 952: 941: 934: 932: 930: 925: 922: 921:power divider 918: 912: 910: 909:isolated port 906: 896: 891: 883: 881: 877: 875: 871: 866: 864: 860: 851: 847: 842: 837: 829: 824: 822: 820: 816: 812: 808: 800: 798: 796: 792: 787: 785: 780: 771: 769: 768:is required. 767: 763: 759: 758:Hilbert curve 755: 751: 747: 742: 740: 736: 731: 727: 723: 719: 715: 706: 701: 693: 691: 689: 685: 680: 674: 666: 664: 659: 651: 649: 647: 643: 639: 631: 629: 626: 622: 618: 609: 605: 600: 593: 591: 589: 585: 580: 578: 571:Coupled lines 570: 568: 566: 557: 553: 551: 544: 536: 534: 528: 526: 523: 518: 510: 508: 505: 501: 494: 489: 484: 476: 474: 472: 468: 464: 460: 456: 450: 442: 440: 438: 434: 430: 426: 419: 414: 407: 405: 403: 400: 396: 392: 388: 384: 380: 371: 366: 364: 362: 357: 348: 346: 344: 340: 336: 332: 328: 324: 319: 317: 311: 309: 305: 301: 297: 289: 284: 277: 275: 273: 269: 265: 261: 257: 253: 250:(PCBs) using 249: 240: 234: 229: 223: 219: 215: 211: 207: 203: 199: 195: 191: 187: 183: 179: 175: 167: 162: 158: 153: 149: 147: 143: 139: 135: 130: 128: 124: 120: 116: 115:coupled lines 112: 108: 103: 101: 97: 93: 89: 85: 81: 77: 74:cheaply as a 71: 69: 65: 61: 57: 53: 49: 45: 41: 34: 30: 26: 21: 3425: 3411: 3396: 3381: 3374: 3355: 3340: 3333: 3314: 3299: 3292: 3281: 3262: 3240: 3233: 3214: 3199: 3184: 3169: 3154: 3139: 3124: 3117: 3106: 3060: 3045: 3030: 3015: 3000: 2993: 2974: 2959: 2944: 2937: 2918: 2903: 2888: 2873: 2861:, IET, 2001 2858: 2843: 2828: 2813: 2801:, IET, 2000 2798: 2782: 2766: 2751: 2736: 2735:Gupta, S K, 2729: 2710: 2695: 2680: 2665: 2650: 2643: 2624: 2609: 2594: 2579: 2564: 2557: 2538: 2523: 2516: 2509: 2494: 2479: 2464: 2449: 2434: 2427: 2408: 2402:Bibliography 2391: 2382: 2364: 2355: 2346: 2337: 2328: 2319: 2310: 2301: 2293: 2288: 2279: 2270: 2261: 2243: 2234: 2225: 2216: 2207: 2202:Borden, p. 3 2198: 2178: 2170: 2165:Mason (1961) 2161: 2156:Mason (1930) 2152: 2143: 2134: 2125: 2116: 2107: 2098: 2089: 2078: 2067: 2049: 2031: 2022: 2013: 2004: 1995: 1986: 1977: 1968: 1959: 1950: 1941: 1923: 1914: 1905: 1884: 1863: 1842: 1828: 1820: 1812: 1807: 1798: 1790: 1785: 1777: 1772: 1764: 1759: 1751: 1746: 1738: 1733: 1725: 1720: 1712: 1707: 1696: 1685: 1676: 1667: 1658: 1649: 1640: 1623: 1616: 1608: 1599: 1581: 1570: 1566:Mason (1961) 1553: 1544: 1535: 1520: 1509: 1493: 1482: 1468: 1460: 1451: 1442: 1433: 1422: 1411: 1403: 1398: 1357: 1342: 1330: 1314: 1301:World War II 1298: 1287: 1264: 1230:, where the 1218: 1195: 1170: 1067:S-parameters 1060: 1050:, where the 1045: 1029: 980: 976:interference 963: 955: 946: 926: 920: 916: 913: 908: 905:coupled port 904: 901: 878: 867: 855: 811:terminations 804: 791:horn antenna 788: 775: 743: 738: 735:pre-fractals 734: 711: 676: 661: 635: 613: 581: 574: 562: 546: 532: 514: 497: 452: 425:Coaxial line 423: 391:Lecher lines 379:twisted pair 375: 352: 320: 312: 293: 241: 171: 142:broadcasting 134:World War II 131: 104: 72: 64:transformers 39: 38: 3298:Roer, T G, 2510:Electronics 1855:Lee, p. 100 1699:, pp. 12–14 1653:Lee, p. 787 1471:, pp. 45–46 1425:, pp. 35–36 1406:, pp. 35–37 1370:, a set of 1206:transistors 1009:Circulators 992:attenuators 964:hybrid ring 852:stub filter 844:Microstrip 807:attenuators 779:transitions 762:Peano curve 726:out-of-band 712:The use of 190:capacitance 127:circulators 3432:Categories 3420:9533070838 3405:1420036866 3390:0471715824 3364:0849389518 3349:0852268580 3323:8121935377 3308:1461525004 3271:9533072474 3249:0080533531 3223:0471398209 3208:364232861X 3193:0471478733 3178:0471720607 3163:1596932430 3069:1608072061 3054:0080492053 3039:3662043254 3024:0849302692 3009:1608075834 2983:0521835267 2968:0815516134 2953:1608078329 2927:0471089192 2912:1107081971 2897:0123746965 2882:1420041312 2867:0852967772 2852:0471464201 2837:0471221651 2822:1608074897 2807:0852967942 2760:0890061572 2745:8187224754 2719:1107170273 2704:1608075354 2674:0932764061 2659:1118936191 2633:160807756X 2618:1461243386 2603:9814366064 2588:0470020458 2573:047144474X 2547:1420069004 2532:8122400523 2503:3319434705 2488:8184317220 2473:1608077128 2458:0470462310 2443:981445124X 2417:0470036958 1615:Magnusson 1492:Magnusson 1391:References 1372:transforms 1353:Microstrip 1275:dispersion 1243:See also: 1214:parasitics 1063:reciprocal 1056:Gunn diode 1026:Circulator 1005:networks. 819:thick-film 722:multi-band 698:See also: 686:and lower 588:resonators 511:Mechanical 504:resonators 455:microstrip 429:dielectric 402:feed lines 343:ideal line 222:wavelength 194:inductance 56:capacitors 33:microstrip 3148:830829462 3133:0026-2897 2292:Matthaei 2265:Ahn, p. 3 2081:, pp. 6–7 1789:Janković 1776:Janković 1763:Janković 1750:Janković 1737:Janković 1711:Janković 1695:Doumanis 1467:Doumanis 1421:Vendelin 1402:Vendelin 1345:stripline 1337:Leo Young 1326:magic tee 1322:Bell Labs 1305:broadcast 1294:Bell Labs 1210:substrate 1071:symmetric 929:bandwidth 846:band-pass 815:thin-film 809:and line 718:wide-band 642:Resonance 477:Waveguide 463:stripline 397:used for 395:twin-lead 308:prototype 233:microwave 218:frequency 202:capacitor 107:resonator 88:waveguide 68:microwave 60:inductors 2177:Johnson 2077:Lacomme 1793:, p. 196 1780:, p. 196 1767:, p. 196 1741:, p. 191 1728:, p. 237 1724:Ramadan 1715:, p. 197 1626:, p. 433 1619:, p. 199 1569:Johnson 1496:, p. 240 1041:duplexer 1037:isolator 857:include 850:low-pass 786:design. 694:Fractals 608:duplexer 577:coupling 393:and the 206:inductor 198:resistor 2792:6884353 2775:3388033 2689:2205252 1512:, p. 73 1299:Before 1239:History 1190:biasing 1033:ferrite 935:Hybrids 859:fractal 766:cascade 714:fractal 690:bands. 467:finline 408:Coaxial 399:antenna 361:quality 244:500 MHz 119:filters 84:coaxial 52:passive 3418: 3403: 3388: 3362: 3347: 3321: 3306: 3269: 3247: 3221: 3206: 3191: 3176: 3161: 3146: 3131: 3067: 3052: 3037: 3022: 3007: 2981: 2966: 2951: 2925: 2910: 2895: 2880: 2865: 2850: 2835: 2820: 2805: 2790: 2773: 2758: 2743: 2717: 2702: 2687: 2672: 2657: 2631: 2616: 2601: 2586: 2571: 2545: 2530: 2501: 2486: 2471: 2456: 2441: 2415: 2296:(1964) 2294:et al. 2179:et al. 2079:et al. 1829:et al. 1813:et al. 1791:et al. 1778:et al. 1765:et al. 1752:et al. 1739:et al. 1726:et al. 1713:et al. 1697:et al. 1624:et al. 1617:et al. 1571:et al. 1510:et al. 1494:et al. 1469:et al. 1423:et al. 1404:et al. 1251:, and 1202:diodes 998:, and 956:hybrid 760:, and 443:Planar 125:, and 98:, and 62:, and 1827:Garg 1811:Garg 1622:Garg 1508:Chen 1054:of a 1019:1 GHz 772:Taper 679:helix 500:modes 437:ports 367:Media 237:1 GHz 214:phase 138:radar 111:stubs 92:radar 3416:ISBN 3401:ISBN 3386:ISBN 3360:ISBN 3345:ISBN 3319:ISBN 3304:ISBN 3267:ISBN 3245:ISBN 3219:ISBN 3204:ISBN 3189:ISBN 3174:ISBN 3159:ISBN 3144:OCLC 3129:ISSN 3065:ISBN 3050:ISBN 3035:ISBN 3020:ISBN 3005:ISBN 2979:ISBN 2964:ISBN 2949:ISBN 2923:ISBN 2908:ISBN 2893:ISBN 2878:ISBN 2863:ISBN 2848:ISBN 2833:ISBN 2818:ISBN 2803:ISBN 2788:OCLC 2771:OCLC 2756:ISBN 2741:ISBN 2715:ISBN 2700:ISBN 2685:OCLC 2670:ISBN 2655:ISBN 2629:ISBN 2614:ISBN 2599:ISBN 2584:ISBN 2569:ISBN 2543:ISBN 2528:ISBN 2499:ISBN 2484:ISBN 2469:ISBN 2454:ISBN 2439:ISBN 2413:ISBN 1232:gain 1224:FETs 1003:feed 793:and 720:and 621:dual 537:Stub 226:300 192:and 86:and 42:are 1347:by 966:or 876:). 817:or 737:or 688:UHF 684:VHF 602:An 385:on 333:of 228:MHz 204:or 184:of 3434:: 3373:, 3332:, 3291:, 3280:, 3232:, 3116:, 3105:, 2992:, 2936:, 2728:, 2642:, 2556:, 2426:, 1362:' 1247:, 1216:. 1204:, 994:, 990:, 986:, 951:dB 949:3 865:. 756:, 752:, 748:, 636:A 491:A 465:, 404:. 258:. 239:. 200:, 188:, 155:A 129:. 121:, 113:, 102:. 94:, 58:, 23:A 3422:. 3407:. 3392:. 3366:. 3351:. 3325:. 3310:. 3273:. 3251:. 3225:. 3210:. 3195:. 3180:. 3165:. 3150:. 3135:. 3071:. 3056:. 3041:. 3026:. 3011:. 2985:. 2970:. 2955:. 2929:. 2914:. 2899:. 2884:. 2869:. 2854:. 2839:. 2824:. 2809:. 2794:. 2777:. 2762:. 2747:. 2721:. 2706:. 2691:. 2676:. 2661:. 2635:. 2620:. 2605:. 2590:. 2575:. 2549:. 2534:. 2505:. 2490:. 2475:. 2460:. 2445:. 2419:. 1154:) 1148:0 1143:1 1138:0 1131:0 1126:0 1121:1 1114:1 1109:0 1104:0 1098:( 1093:= 1090:] 1087:S 1084:[ 35:.

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Knowledge (XXG)

Distributed-element circuit

Index